Skip to main content

Full text of "DTIC ADA040475: Titan IIIC Guidance with the Carousel VB Inertial Guidance System. Reissue A"

See other formats


REPORT SAMSO- TR-75-297 
{SUPERSEDES TR-73-220) 



!> 

- I I 

< 

Q 

<£ 


Titan IIIC Guidance with the Carousel VB 
Inertial Guidance System 


T. E. DARNAUD and J.B.SHAUL 


Guidance and Control Division 
Engineering Science Operations 
The Aerospace Corporation 
El Segundo, Calif. 90245 



\ 

i 


15 July 1975 





Prepared for 

SPACE AND MISSILE SYSTEMS ORGANIZATION 
AIR FORCE SYSTEMS COMMAND 
Los Angeles Air Force Station 
P.0. Box 92960, Worldway Postal Center 
Los Angeles, Calif. 90009 




Approved 


1 / 

/J. R. Allder, Director 
Guidance and Flight Dynamics 
Subdivision 

Guidance and Control Division 
Engineering Science Operations 


C j). 

D7 Baxter, Director 
Titan III Directorate 
Vehicle Systems Division 
Systems Engineering Operations 


Publication of this report does not constitute Air Force approval of 
the report's findings or conclusions. It is published only for the exchange 
and stimulation of ideas. 

^RICHARD E. WOLFSBERGER, Cornel, USAF 
Assistant Program Director 
Expendable Launch Vehicles SPO 

This report supersedes and replaces TR-0073(34 1 3 -02)- 1 , dated 29 
November 1972. 



UNCLASSIFIED 


SECURITY CLASSIFICATION OF THIS PAGE fJWi*. Data Bntatad) 


REPORT DOCUMENTATION PAGE 


SA MS0.¥TR - 75-297 


READ INSTRUCTIONS 
BEFORE COMPLETING FORM 


*• COvT ACCESSION NO. S. RECIPIENT'S CATALOG NUMBER 


5. TYPe OP REPORT A PERIOD COVEREO 



7. AUTMORfsJ 


) T. E. yfearnau dRMB J. RT^haul) 


PggTTW: 


ACT OR GRANT NUMBERf.J 


9. PERFORMING ORGANIZATION NAME AND ADDRESS 

The Aerospace Corporation 
El Segundo, Calif. 90245 



10. PROGRAM ELEMENT. PROJECT. TASK 
AREA * WORK UNIT NUMBERS 




!S 


11. CONTROLLING OFFICE NAME AND ADDRESS 

Space and Missile Systems Organization 
Los Angeles Air Force Station — ■ 

Los Angeles, Calif. 90045 £ 


14. MONITORING AGENCY NAME A AODRESS (II dt Hat ant I mm Control I Inf Ollleo) IS. SECURITY CLASS, (ol Ihlo tapoH) 

Unclassified 

IS*. DECLASSIFICATION/DOWNGRADING 
SCHEDULE 


16. distribution statement (oi thio Raport) 


Approved for public release; distribution unlimited 


17. DISTRIBUTION STATEMENT (ol Iho abotracl antarad In Block 30, II dl liar an I from Ha port) 



>9. KEY WOROS ( Continue on nvirat «id« If nKMitry and Identity by bloc* nunbar) 


Guidance Equations 
Inertial Navigation ~ 

Carousel VB -3 

, Rocket Steering Law 
\ Vehicle-Borne Computer Program 


BSTRACT (Contlnua on ravmraa alda It ttacaaaary and tdantlty by block numb«0 

The Titan II^C Standard Space Launch Vehicle, starting with Vehicle 26, 
will use the Carousel VB Inertial Guidance System for navigation, guidance, 
and digital flight controls. ''This guidance system consists of a Carousel VB 
Inertial Measurement Unit (1MU) and a MAGIC 352 Missile Guidance Com- 
puter (MGC), both manufactured by Delco Electronics, General Motors 
Corporation, The C-VB IMU is a modification of the C-IVB inertial navi- 
gator currently in airline service as a primary navigation system. — ^ 


00 ruKM 1473 

irACSiMlLCl 


a 


UNCLASSIFIED 

SECURITY CL AMI F 1C AT ION OF THIS RAOC !«*•« DM* «**•»•« 








SB®®®*** 


SECURITY CLASSIFICATION or THIS RA<3E(Trh«n Dim Bill) 





ABSTRACT (Conttnumd) 


' A unique feature of the Carousel IMU is that two of the three gyro/ 
accelerometer sets are revolved at 1 rpm with respect to the stable 
platform. The effect is to partially cancel certain instrument errors 
associated with the revolving instruments. Conversion of this instrument 
from commercial airline navigation service to guidance and control of 
the rocket boost vehicle presented a guidance software design task that 
is the subject of this paper. 


UNCLASSIFIED 

SECURITY CLASSIFICATION OF TNI* RAOCfOM* Oat* 






PREFACE 


The guidance equations derived in this paper were developed by 
the authors in a joint effort with D. L. Kleinbub and A. C. Liang of The 
Aerospace Corporation. 

A summary of the first issue of this report was presented at the 
Sixth Hawaii International Conference on System Sciences January 11, 1973 
and was published in the proceedings. 



CONTENTS 


m 








:\ m 



m 

fM 




:ni poduction 


MISSION SUMMARY AND VEHICLE DESCRIPTION 


General 


Vehicle Description . . . . 

1 . Stage 0 

2. Core Stages I and II 


Rage III 


Control Module 


Mission Description (Flight Plan VIU 
1 . Ascent to Parking Orbit 

2. Parking Orbit 

3. Stage HI First Burn 


4. Transfer Orbit 

5. Final Orbit Injection 

6. Paylcad Separation 

GUIDANCE, STEERING, AND NAVIGATION 


Gene ral 


Guidance 


Thrust Act deration Prediction 


Pitch Stee ring 
Yaw Steering 


Estimate of Time to Go 


5. Integral Control . . . . 

6. Maneuvering Equations 

Navigation 


Accelerometer Resolution and 
Compensation 


2. Gyro Drift Compensation 


f PRECEDING PIGS BLANK-HOT FUME] 

' T* 


- 3 - 




CONTENTS (Continued) 


3. Transformation of Velocity to Inertia] 

x, y, 7, Coordinate System 

4. Vehicle Inertial Position . 

5. Gravity Computations 

6. Vehicle Inertial Velocity 

D. Attitude Errors . 

1 . Synchros 

2. Attitude Error Computations 

REFERENCES 


FIGURES 


1. Titan I1IC External Prolile . ... . 

2. Typical Titan IIIC Mission Profile 

3. Carousel VB IMU Geometry 

4 Navigation Block Diagram 

TABLE 

1 . Carousel VB IGS Characteristics . - 


SECTION I 


INTRODUCTION 


Referent el is a complete specification of the guidance equations 
for the Titan HIC launch vehicle. It includes all information necessary for 
programming the launch vehicle guidance computer. Reference 1 does not, 
however, include any discussion of the guidance algorithms or the derivation 
of equations. The equations specified by Ref. 1 and discussed in this 
document are applicable to SSLV C-26 and subsequent vehicles. 

Section II, a very general description of a typical Flight Plan VII 
mission and of the Titan IIIC vehicle, is included as an aid to understanding 
the inflight equations, some aspects of which arc vehicle- and/or rnission- 
peculia r. 

Section III is a discussion of the Titan IIIC navigation, guidance, 
and steering. 


- 5 - 





SECTION II 


MISSION SUMMARY AND VEHICLE DESCRIPTION 


A. GENERAL 

The Flight Plan VII mission consists of injecting a payload(s) into a 
synchronous or near- synchronous equatorial orbit using the Titan IIIC SSLV. 
The combined vehicle Stages II and III, together with the payload, are 
injected directly into an e'liptical parking orbit having a perigee of approxi- 
mately 80 nmi and an apogee of approximately 235 nmi. At the first 
descending or ascending node, depending on mission options, a first burn of 
Stage III produces an elliptical orbit with an apogee of approximately 19,323 
nmi and an orbital inclination . educed by approximately 2-1/4 deg. At apogee, 
a second Stage III burn produces the desired synchronous (or near synchronous) 
equatorial orbit. 

B ■ VEHICLE DESCRIPTION 

The Titan IIIC launch vehicle shown in Fig. 1 consists of a solid 
motor stage and three liquid engine stages plus a control module. The solid 
motor stage is referred to as Stage 0, and the three liquid engine stages are 
referred to as Core Stages I c II, and Stage III. The control module and 
Stage III are sometimes collecti vely called the transtage. The stages are 
briefly^ described in the following paragraphs. 

1. STAGE 0 

Stage 0 consists of two solid propellant rocket motors positioned 
parallel to the standard core in the yaw plane. Thrust vector control is 
accomplished by injecting oxidizer (N^O^), pressurized by gaseous nitrogen 
(N^), into any of four quadrants of the nozzles. 


PRECuDINO PAJS BLANK-HOT PIIirsB" " 


- 7 - 




1 


PAYLOAD FAIRING- 


STATION 


FAIRING AND PAYLOAD- 
INTERFACE 

STAGE il AND STAGE III- 
SEPAR ATION PLANE 


-CONTROL MODULE 


STAGE II 


STAGE I A.'O STAGE II- 
SEPAR ATION ’’LANE 


/' 7 °h 

' l\ ]/ — 24. 00 R 

PK\ “ ' - 360 - 15 


| \ V-,z 


125.80 

(typ each segment) 


STAGE I — 


-12.00 R 


STAGE 0 (2 SRMs)- 


5 SEGMENT SOLID- 



6 OFFSET v 


r T "1 14- 


-TVC TANK 




H \ 


2° OFF SET — 

S^O’ GIMBAL-Jf-^n 5° 3 O' GIMBAL p 

(max) 




—1213. 10 
—1224. 3 1 
1240.68 
— 1274. 12 

1304.25 

—1326.00 

—1348. 10 (nozzle exit) 
- 1375.79 


ROLL AXIS 


HEATSHIELD —r'/ b.L.O. 

THRUST CHAMBER—' | 
COVERS ! 


-117° 24' 
(typ) 


85.00 R- 

(typ) 


<fc. YAW AXIS^jargex 


126. 51 

(typ) I 


-W. L. 60 


-<E PITCH AXIS 


Figure 1. Titan IIIC External Profile 


- 8 - 




z. 


CORE STAGES I AND II 


Stages I and II are powered, respectively, by dual and single *hrust 
chamber, turbopump-fed, liquid propulsion systems. These systems utilize 
storable propellants, a 50:50 mixture of hydrazine and unsymmetrical 
dimethylhydrazine (UDMH) for fuel and nitrogen tetroxide (N^O^) for oxidizer. 

3. STAGE III 

The Stage III propulsion system consists of two pressure-fed engines. 
Stage III also contains an attitude control system (ACS), which provides atti- 
tude control during coast, ullage control prior to main engine burns, and 
velocity additions for vernier phases and for satellite ejections. The pro- 
pellants used for the main propulsion system are the same as for Stages I 
and II. The attitude control system employs hydrazine monopropellant 
engines. 

4. CONTROL MODULE 

The control module contains the major elements of the flight control, 
inertial guidance, electrical, telemetry, tracking, and flight safety systems. 
The control module is attached to the forward end of Stage III and remains 
attached throughout flight. 

C. MISSION DESCRIPTION (FLIGHT PLAN VII) 

The following is a brief mission description of a typical Flight Plan 
VII. All values given are only approximate, and any discussion of the Flight 
Plan that does not contribute to understanding the guidance equations has been 
intentionally omitted. A complete description of a Flight Plan VII mission 
is given in Refs. 2 and 3. Figure 2 is an illustration of the Flight Plan. 

1. ASCENT TO PARKING ORBIT 

The vehicle is launched vertically from Air Force Eastern Test 
Range (ETR) Pad 41 or 40; shortly after liftoff, it is rolled from a pad azimuth 
of 100. 2 deg to a flight azimuth of 93 deg. Following the roll maneuver, an 
open-loop pitchover, with load relief during the region of max Q, is performed 
for the remainder of Stage 0 burn. Stages I and II are closed-loop guided; 


- 9 - 



ss 











•H 



the engines burn for approximately 149 and 207 sec, respectively, and the 
stages burn to propellant exhaustion. At Stage II propellant depletion, the 
vehicle is nominally in an elliptical orbit 80 X 235 nmi. Further, in the case 
of a low performing vehicle, the minimum orbit is 80 X 95 nmi. 

2. PARKING ORBIT 

Immediately after injection into the parking orbit, the vehicle attitude 
is adjusted to the standard orbital orientation: vehicle longitudinal axis 
normal to the geocentric radius vector, the nose of the vehicle pointed approxi- 
mately along the velocity vector, and the vehicle pitch plane coincident with 
the orbital plane. 

After coasting in the parking orbit to near the f-rst descending or 
ascending node, the vehicle is reoriented in yaw as a startup attitude for the 
first burn of the transtage. Concurrently, a roll maneuver is performed 
so that the vehicle telemetry antenna is pointed toward the desired ground 
tracking station upon completion of the reorientation maneuver. 

3. STAGE III FIRST BURN 

Close to the equatorial crossing, the Stage III engines are ignited 
and burn for approximately 300 sec. The vehicle is injected into an elliptical 
orbit with an apogee of 19, 323 nmi. The first burn includes an orbital plane 
change maneuver of approximately 2. 25 deg. If required, the ACS of 
Stage III can adjust the orbital parameters by a vernier velocity addition. 
Nominally, shutdown of the Stage III engines is controlled so that a 6- sec 
ACS vernier phase is required. 

4. TRANSFER ORBIT 

During the transfer orbit, the transtage performs certain maneuvers 
to meet thermal control requirements. Currently, three options axe being 
considered for thermal maneuver. The first, called rotisserie, is an oscilla- 
tory roll maneuver of approximately ±115 deg at a rate of 1 deg/sec, with a 


- 11 - 


f -min dwell time at cat. h cxlremo position. In the sei ond maneuver, 
toasting. th< vehicle is simply turned hack and forth in space at widely- 
spaced time intervals, l’he third is a continuous roll maneuver between 1 
and 2 deg/ set. In all three options, the relationship of the sun vector to the 
vehicle or spacecraft axes is specified by the payload thermal requirements. 
In addition to the thermal maneuvers, the vehicle is oriented several times 
during the transfer orbit tc an attitude that permits reliable reception of 
telemetry. 

Finally, shortly before reignition of the Stage III engines, the 
vohic le is reoriented to a startup attitude for the second burn, which also 
points the telemetry antenna earthward. 

n. FINAL CRB IT INJECTION 

The second Stage III burn, of approximately 104- sec duration, 
injects the payload and /ehicle into a circular orbit at 19,323-nmi attitude, 
with a near zero-deg inclination. 

6. PAYLOAD SEPARATION 

After the second Stage III shutdown, the vehicle is reoriented for 
the satellite separation phase. This orientation can vary depending on the 
selected mission. After sufficient time for stabilization at the desired atti- 
tude, the ACS is switched (o the payload release coast mode; after sufficient 
time for stabilization in this mode, the payloaa is released. At this point, 
the Titan IIIC mission can be ended with a transtage shutdown sequence. 
Alternatively, the equations provide an option for performing a short 
propellant settling burn, another payload release sequence, and, finally, the 
transtage shutdown sequence. 

In both of these payload release options, the equations issue a 
variety of required discretes and can perform multiple reorientations as 
specified by payload requirements. 


- 12 - 


SECTION III 


GUIDANCE, STEERING, AND NAVIGATION 


A. GENERAL 

The vehicleborne digital computer program is divided into three 
distinct sections: digital flight control equations, ground equations, and 

guidance equations. This document treats only the guidance equations; the 
other programs are mentioned only when they interface with the guidance 
equations. The carousel sensor and gimbal geometry is given in Fig. 3, 
while principal computer and sensor characteristics are tabulated in Table 1. 
Figure 3 and Table 1 are modifications of data taken from Ref. 9. 

The guidance equations are organized into various subprograms. 

A main control or executive program controls logical routing to appropriate 
calculations , depending on the phase of flight and/or significant flight events. 
The navigation equations accept inertial measurement unit (IMU) data and 
compute current vehicle position and velocity. Booster steering equations 
calculate open-loop steering for Stage 0. Powered flight equations provide 
vehicle steering as a function of navigational quantities and desired end con- 
ditions. Several coast phase subprograms provide computations for desired 
vehicle attitudes and event initiations during nonpowered flight. Finally, the 
specific flight plan and the multistage Titan 1IIC vehicle necessitate numerous 
subprograms to repeatedly reinitialize the guidance equations for each phase 
of flight and to issue discretes to control events such as engine shutdown, 
engine ignition, and stage separation. The following paragraphs describe 
the guidance philosophy and overall logic flow of the guidance equations. 

B. GUIDANCE 

The guidance philosophy governing the guidance equations presented 
in this document is commonly termed 'explicit" guidance. Generally, an 


- 13 - 





v> 

H 

D. 

or 

U 

tO 

m 

D 

W 


Ul .. 
I -J 


v> 

a 

o 


a 

uj 

z 


o 

Ui 

> 


o 

o 



o 

u. 


< 
h- 

z 

UJ 
UJ - 
I £ 
h* O 


_J UJ 
-J H 



JO 

X 

< 


0 
or 

v> 

1 5 

5 

K 


? = £ 


I 

U 

J~ 


t o 

x 

< 

i 

< 

>- 


J2 

X 

< 

* 

< 

>- 


^ 


*5 

‘5 

s 

*3 

g 


Oi 

u 

o 

V? 



UJ 

UJ 

UJ 

UJ 

UJ 


-J 

-J 

-J 


-J 


u 

X 

u 

X 

o 

X 

u 

X 

o 

X 


UJ 

UJ 

UJ 

UJ 

UJ 

>> 

> 

> 

> 

> 

> 

ft 


cr 

O 


•- t-P ^ >* 

o td o 


£ 

o 

a 

D 

2 


CQ 

> 


u* 

X 

UJ 


H 

Z 

UJ 

a 

O 

o 

z 


o 

a: 

x 

¥ ° 
? * 
>" UJ 
” N 

x“>- 


o 
z 
< 

cn 05 
^ UJ 

u 


a: 

< 

t/> 


V) 

UJ 

U 

z 


t/> 

J- 

w w 

a o 
*- z 

z < 

to 

z < 

— ffl 

ui o 

S UJ 

o x 


~J 

? h a 

a h 

-J 

o 

< D UI 
i/> <0 _l 

^ f 

U. 

UJ 

- O 

Z J *- 

uj O 
x 

I 

O -j z 

o 

J— 

i= < s 

u 
u_ h 

J 

< J 1 . 

t- ^ ,J 


< 

►- 

,? It w 
y a or 

§ 

— Q. 

O' 

a: < ui 

t~ « 

UJ 

z 

0 w £ 

< UI 

f- a: 

2 -r 

Q 

lit O UI 

y 2 

O 

X Z D 

5 ~ 

K 

t- < Q 

O 2 i 

< 

• 

• 


>- 

or 

< 

a: 

h 

m 

a 

< 


a 

z 

o 

<r 

>- 

o 

o 

z 

< 

(X 

UJ 


a 

UJ 


0 
u 

< 

UJ 

1 


UJ 

(X 


o 

o 

z 

< vi 

< w 
5 x 

< « 


a; 

0) 

d 

o 


aJ 

O 


CO 

<b 

U 

r-* 

&& 
■ H 

h 


- 14 - 



explicit guidance scheme is one in which an explicit solution to the powered 
flight dynamics is computed in flight and the desired end result of guidam e 
is explicitly defined in the guidance equations. The Flight Plan VII guid- 
ance equations, deoenclmg upon the flight phase, define different desired 
end conditions. The burns of Core Stages I and II are planned for a specific 
position magnitude, velocity magnitude, radial velocity, and preplanned 
orbit planes. The end result of guidance for the first burn of Stage III 
(transtage) is injection onto a transfer ellipse whose apogee, semi-major 
axis orientation, and orbital plane are specified. Finally, during the second 
burn, the transtage guides to a near-circular orbit at the transfer ellipse 
apogee radius and the orbit piano is specified (approximately equatorial). 

References 4, 5, and 6 all contain general discussions of explicit 
guidance algorithms. 

1. THRUST ACCELERATION PREDICTION 

A basic requirement of the guidance equations presented here is a 
prediction of vehicle performam c, particularly of thrust acceleration. For 
this report, "thrust acceleration" is used as an abbreviation for "accelera- 
tion (of the vehicle) due to thrust. " The following paragraphs derive a 
time-dependent expression for thrust acceleration. 

Consider the variable T , defined as 

r 

= W/W (1) 

whe re 

W - total weight of vehicle at any time 
W • rate of change of vehicle weight 

T^ can be interpreted as the time remaining to zero vehicle weight (mass 
intercept), if a constant W is assumed. 


- 16 - 




Since specific impulse is 


1 = Thrust/ W 

sp 


and thrust acceleration is 


A th -- Thrust/ (W/g) 
by solving for thrust and weight as 


( 2 ) 


( 3 ) 


W = T W 


Thrust = W1 


sp 


( 4 ) 


one can express A„,, as 
1 H 


A—u - gl /T 
TH 6 sp r 


( 5 ) 


Further, since the effective exhaust velocity is given as 


V = gl 
e 6 sp 


( 6 ) 


then it follows that 


A TH = V e /T r 


( 7 ) 


Finally, since T^ is a linear function of time, then A^,^ as a 


function of time t can be expressed as 


A TH' 1 ' - V e /(T rO* t) 


( 8 ) 


where T r Q = value of T r computed at t = 0. 


- 17 - 




2. PITCH STEERING 

I" polar coordinates , the two-body equations of motion for a vehicle 
in a central for. c field gives the radial acceleration as: 


^ ^THr ” 


wher< 


^THr ~ rdciia l thrust acceleration 


L. n 

w *'■ centrifugal acceleration 
2 

|i / R ~ acceleration due to gravity 

If a time-dependent guidance steering law for the radial thrust 
pointing d 1 rection is assumed to be 


A THr /A TH T A ' Bli f WR 2 - ci 2 R)/A th ] 
then, upon substituting for A^, one finds that Eq. (9) becomes 


RMA-t Bt >A TH (11) 

This steering law, the familiar "linear sine" law, has been applied 
to numerous guidance programs. It has the advantages of simplicity and 
very near fuel optimality for current ro''k of - r '-.'>'.v c 'n c ,i u. 

Since explicit guidance equations require a prediction of vehicle 
motion, it is desirable to time-integrate Eq. (1 1} twice to estimate future 
vehicle radial velocity and position. However, before these integrations 
are attempted, Eq. (11) can be simplified to reduce integration complexities. 


- 18 - 


The* linear assumption of T can be written as 


t 



( 12 ; 


Substituting Eq. (12) into Eq. (11) yields 


R = (A + BT r0 - BT r )(A TH ) 


(13: 


And, after some algebraic manipulation, 


R = -BT A th + (A + BT rQ )(A TH ) 


= -BV + (A i BT r ) (A_ u ) 
e rO TH 


(14) 


In Eq. (14), the quantities A,B,V o , and are assumed constant; 


therefore, Eq. (14) can be rewritten as 


R = A' + B A 

r r T H 


(1 5> 


Time integration of Eq. (15) from a given time (t = 0) to cutoff (t - T ) yields 


V rr V 


r T r T 

+ / 8 A' dt + B / 8 A„ u dt 

r Jn r J 0 TH 


( 16 ) 


/•T ft 

f, % *> 


R, = R + V T + 
f r g 


dt + B *fn 8 f A TH dS 

JO Jo 


dt 


- 19 - 



where 


&S* ■' s 

M 4 


WM 


|g|| 

il 


V = present radial velocity 
r 1 


V r - radial velocity at cutoff 
rf 


iir 


%k4. 


m 


R = present ra fiai position 
= radial position at cutoff 

The integrals of thrust acceleration in Eq. (16) can be evaluated 
using the time-dependent expression for A^^, given in Eq. ( 1 Z) : 


nil 


h M 


r 

/ s 

Jo 


A.dt - V 
Th g 


A ds di - V T - V T 4 V T 
IH g g g rO eg 


in these integrals, it is assumed that V is known. A derivation of the 
b g 

second integral (a^) can be found in practically any discussion of an explicit 
guidance technique, such as R .. 4. 

To solve for the steering coefficients A^ and B f> one must substitute 
the values for the integrals in Eq. (17) into Eos. (16): 


V c - V 4 A' T + B V 
rf r r g r g 


R, R + V T 4 A' (T 2 /2) 4 B (a, ,) 
I r g r g r 1 2 




it 



(19) 


Solving for yields 


B = R, - R - (T / 2)( V , + V ) 
r f g ” rf r 


T ro - <y 2 > 


+ V T 
e g 


and for A', 
r 


A' = (1/T )(V , - V - V B ) 
r g rf r g r 


( 20 ) 


Finally, the expression for the desired radial acceleration A^ .s 
obtained by substitution into Eq. (15): 


A 

r 


R = (1 /T )(V , - V 
g rf r 


- V B ) + B A, 
g r' r 


r H 


( 21 ) 


3. YAW STEERING 

The yaw steering is derived in a manner analogous to the pitch 

steering scheme. In the yaw case, a desired orbital plane is specified uy a 

vector normal to the plane. Normal error parameters R and can be 

computed as the dot product of vehicle inertial position and velocity with the 

specified normal vector. Since the desired value of both error parameters 

is zero at cutoff, the following equations can be used to determine the yaw 

steering coefficients A and B : 
f n n 


0 = V + A T + B V 
n n g n g 


0 = R + V T + A T / 2 + B a, _ 
n n g n g n 1 2 


( 22 ) 


- 21 - 


4. 


ESTIMATE. OF TIME TO C.O 


* .1 


The estimate of time to go is derived from AM, the vehicle angular 
momentum to bo gained, which is given as 




AM = M f - M 


"M 


where 


r<; 


. « 


M^ = desired vehicle angular momentum magnitude at cutoff 
M - present vehicle angular momentum magnitude 




Angular momentum to be gained is also given by the integral equation 


AM = 


where T ^ is the time until rocket engine cutoff. 

To estimate time to go, solve the integral Eq. (24) for T by a 

8 

numerical integration technique. The magnitude of angular momentum is 
given as 


M = Rv 




v^ - velocity in the tangential direction 


R = position magnitude 


Per unit mass 






Differentiating Eq. (25) produces 


M = Rv t + Rv^ 


(26) 


Another expression based on orbital and tangential thrust acceleration is 


\ = -\ R/R + A THt (27) 

where is the vehicle thrust acceleration magnitude in the tangential 

direction. Substituting the expression for into Eq. (26) yields for M the 
simple expression 

M = Rv t + R(-v t R/R + A THt ) = RA THt (28) 


The integral of Eq. (24) is evaluated using Simpson's integration 

formula, with the integrand computed at t = 0, t = T , and three equally- 

6 

spaced points in between. 


The first factor in the integrand, position magnitude, is computed at 
the five desired time points by a linear interpolation between the present 
vehicle position magnitude R and the final position magnitude R^; thus, 


R(tj) = R + (j - l)(R f - R)/4 j = i, ...5 (29) 

Thrust acceleration magnitude as a function of time has been derived m 
Paragraph III. B. 1. Thus, it follows that 


a th<V ^ 


V 


J ' 


T r0- - 


J )T / 4 
g 


j = 1. 


(30) 


Where T is tnc time-to-go from the last major cycle decremented by 1 sec 

g 


- 23 - 


The tangential acceleration A,p^ is obtained from a square root as 

, 1/2 


A THt ( y = 


A TH ( V " A THr { V " A THn ( V 


( 31 ) 


AIL tliat remains is to compute the normal and radial thrust accelerations 

A TIIn anc * A THr at Rie re< 5 uire< ^ <- irn e points. 

The present radial components of thrust acceleration have already 

been given as 


A 


THr 


a; + B r A TH - (-p/R 2 + u> 2 R) 


(32) 


and at cutoff as 


A_ ir . - A' + 
THrf r 


B 


- A THf ” + w f B f ) 


(33) 


where the subscript f designates final values at cutoff. The centrifugal 
acceleration terms in Eqs. (32) and (33) can be rewritten, respectively, as 


2 

tjJ 


R = 



(34) 


? 2 
w*R f = M { 



Finally, by linearly interpolating the accelerations due to gravity and centrifu- 
gal force and by employing the value of acceleration magnitude derived pre- 
viously in Eq. (30), one obtains for A^,^^ at the required time points 

A THr (t j ) = A r + B r A TH (t j } " <^ /r2 + m2 / r3 ) 

- (j - l){[(-p/R 2 + M 2 /R f 3 ) - (-p/R 2 + M 2 /R 3 )]/4| 


(35) 


1, ... 5 


- 24 - 



The normal component of thrust acceleration is given simply as 


A THn ( V = A n + B n A TH ( V J = 1 > ■■■ 5 (36) 

By combining the results of Eqs. (29) and (31), one obtains the integrand of 
Eq. (24) at the specified time points 

M(t.) = R(t.)A THt (t.) (37) 

and, using Simpson's integration rule, one can compute an estimate of AM: 

AM' = [M(t } ) + 4M(t 2 ) + 2M(t 3 ) + 4M(t 4 ) + M(t 5 )]T'/l2 (38) 

where AM' is the angular momentum if the initial estimate of time to go is 
assumed. By comparing AM, the true angular momentum to be gained, with 
AM', one can compute an adjustment to the time to go T 

it 

T a = (AM - AM')/M(t 5 ) (39) 


Finally, the time to go becomes 


T 

g 


= T' + T 
g a 


(40) 


With time to go T and the time to mass depletion T , the acceleration 

O 1* 

integral V is computed as 
S 


V 

g 



Iv /(T 
j e ' rO 


t) dt = -V ln(l - T IT J 
e o' r 0 


(41) 


- 25 - 



The guidance scheme described here is iterative. The steering coefficient 
calculations require an estimate of time to go T , while the time-to-go 
calculations presuppose a knowledge of the steering coefficients. Fortu- 
nately, experience has shown that with reasonable startup values and 
appropriate updating between major guidance cycles, the equations described 
here ccnverge quite rapidly for the mission considered. 

5. INTEGRAL CONTROL 

In Paragraphs III. B. 2 and III. B. 3, expression's were derived for 
desired radial and normal thrust accelerations. If these accelerations were 
translated directly into vehicle axis pointing-direction commands, a thrust 
acceleration pointing e**ror would result because of vehicle ard IMU prop- 
erties, such as engine misalignments, vehicle center of gravity effects, 
and IMU gimbal misalignments. Integral control, the method for overcoming 
these pointing errors, is used to measure differences between desired and 
actual guidance- computed accelerations. These differences are numerically 
time-integrated with an appropriate weighting factor for stability reasons, 
i.e. , a digital filter, and finally subtracted from the desired radial and 
normal accelerations. The final steady state effect is to bias the vehicle- 
pointing commands so that the observed misalignments are canceled. 
Preference 7 contains a complete description of integral control. The 
equations are mechanized as 


AA ,r AA n + (C: F6 } - A n J l4Z) 

AA r -AA r + (c F6 ) [(AV r /At) - Aj (43) 


The arrow indicates that the results are stored in the original memory 
location . 


- 26 - 



where 


i 

i 

i 

> 





AA and AA 
n r 


integral control terms 


Cj.,^ integral weighting factor 
A^ - desired normal component of total vehicle acceleration 
A - desired radial component of total vehicle acceleration 


AV /At - guidance-computed normal component of total vehicle 
acceleration 

AV /At guidance - computed radial component of total vehicle 
acceleration 

At 1 -second major cycle interval 


The corrected desired normal and radial equations are then given as 


nc 


= A 


n 


kA 


(44) 


A 

rc 




2 

u> r 


(45) 


Equation (45) includes terms for gravity and centrifugal accelerations, these 
terms were added so that A^, excluding the effect of AA f) would represent 
the desired radial thrust acceleration. 

Dividing Eqs. (44) and (45) by thrust acceleration magnitude yields 
the desired unit vehicle longitudinal axis in the radial, normal, and tangential 
coordinates 


ID = A /A_„ 
&n nc TH 

(46) 

ID = A / A u 
&r rc TH 

(47) 

1/2 


U £t * (* - °*n - °fr) 

(48) 


- 27 - 


Then the unit desired vehicle longitudinal roll axis vector U| is converted 
to inertial coordinates by a matrix multiplication as 


fix" 




i 

C 

in> 

r-f- 

1 


- 

N 


U £n 

U- 




- U £r_ 


( 49 ) 


wiiere N is the matrix that relates tangential, normal, and radial coordinates 
to EC1 (x, y, coordinates. 

The desired unit vehicle pitch axis is arbitrary from a guidance 
standpoint and is selected by the guidance equations to conform to vehicle 
and/or telemetry antenna constraints. Finally, the desired unit yaw axis 
is formed from U^ x to complete a right-hand orthogonal coordinate 
system. 

6. MANEUVERING EQUATIONS 


This section presents a derivation of the maneuvering equations. 

In this paper, maneuvering is considered the application of rate limits to 
the guidance commands (desired vehicle axes) and, if the desired maneuver 
is large, the computation of an efficient maneuver. In addition, the maneuver- 
ing equations calculate values to interpolate between successive major cycle 
commands on a minor cycle basis. The attitude error equations, performed 
on a minor cycle basis, take the output commands from the maneuvering 
equations in the form of desired vehicle axis vectors. The desired vehicle 
axes are compared with the actual present vehicle axes (read from the 1MU 
attitude sensors), and vehicle attitude errors are computed. The flow is 
from guidance equations to maneuvering equations to attitude error equations. 


- 28 - 



The initial calculations made by the maneuvering equations are a 
coordinate conversion from earth centered inertial (ECI) to drifted launch- 
centered inertial (LCI), as follows: 

[rg] t = [0] T [*][cg] 

u £r - [RG] [U $ ] 

5 6 (50) 

U^ = [RG] [U c ] 


where U^ c , U^, and U^ c are the final desired commanded- vehicle roll, 
pitch, and yaw axes in drifted LCI coordinates: [CG] is a coordinate trans- 

formation matrix from ECI to LCI coordinates; [p] is the present compen- 
sable IMU drift matrix; and 4* is a matrix chosen to rearrange the rows of <p 
so as to be compatible with the LCI coordinate system. 

The drifted LCI coordinate system^ is used throughout the following 
equations, since it is the most convenient coordinate system for the guidance 
minor cycle calculations; the IMU attitude readouts are referenced to the 
drifted LCI coordinate system. The drifted LCI system (called a, b, g) can 
also be considered an ideal gimbal angle system. 

Following the coordinate transformation, one can calculate the 
transformation matrix of the desired maneuver: 



u £c ' 

u 4a 

U !c- 

V 

u 4c ‘ 

u r 

M = 

^nc ' 

°4a 

“V 

°na 

°nc- 

U„ 

4a 


_V ' 

u u 

V' 

V 


U y 

4a 


where Uc , U , and U r are the present desired commanded- vehicle roll, 
pitch, and yaw axes. The trace of the BG matrix can be related to the 
commanded maneuver as 


- 29 - 


Trace - 1 - 2 cos 8 


(52) 


o r 


cos 0 (1 - Trace)/2 

where 0 is the magnitude of the maneuver. 

Cos 0 can be tested to determine whether to command a rate- 
limited maneuver. If cos 0 is sufficiently close to 1.0, the maneuver is 
small and the following calculations are performed: 


dU !L = U ^c ' U U 


and 


(53) 


ciU T = U - U 

ol 0c na 


The variables dU^ and dU^j represent the motion of the commanded-body 
axes Ut and U during the next major cycle. If dU f . T and dU T are multi- 
plied by the reciprocal of the number of minor cycles per major cycle, C^q, 
the commanded-body axes can be interpolated on a minor cycle basis. With 


6R 

= dUt T 

cm 


oP 

= dU T 

cm 

nL 


(54) 


computed on a major cycle basis, the following minor cycle calculations 
yield the interpolated axes: 


- 30 - 




cm 


( 55 ) 


U |dm 

^>ldm 


«- u 


<- u 


^dm 

qdm 


+ 6 R 
+ 6P 

cm 


If cos 0 is not close to 1.0, a large maneuver has been commanded 
and rate- limiting is desired. The maneuvering strategy selected consists 
of maneuvering about a single inertially fixed axis at a constant angular 
rate. It can be shown (Euler's theorem) that this single-rotation axis is 
an eigenvector of the maneuver transformation matrix given in Eq. (51). 

The following paragraphs describe an algorithm for computing a command 
“eigenvector" maneuver. 

To solve for an eigenvector, E, of BG, the maneuver transforma- 
tion matrix, consider a skew- symmetric matrix D, calculated as 

D = BG - BG T (56) 


The D matrix has the properties 


[D] 

[E] 

= to] 

(57) 

D. . 
!J 

= 0 

if i = j 

(58) 


From these properties of the D matrix, the following set of 
equations is derived: 


D 12 E 2 + D 13 E 3 " 0 


' D 1 2 E 1 + D 23 E 3 = 0 


-°13 E 1 ' D 23 E 2 " 0 


(59) 


- 31 - 


Solving for the eigenvector in Eqs. (59) yields 


E 1 ' D 23 BG 23 ' BG 32 

E 2 ” " D 1 3 " BG 3 1 BG 1 3 (60) 

E 3 ^ D 1 2 = BG 1 2 ' BG 21 

Alter unitization, the eigenvector is transformed from body 
coordinates to the LCI coordinate system with the following matrix 
multiplication: 


RL = [A] 


UE. 


UE. 


UE 


3 J 


(61) 


where 



U 


UJ 


1 the body axes as columns. 


The magnitude of the eigenvector computed in Eq. (60) is propor- 
tional to the sine of the maneuver angle 0. There is a singularity wherein 
the magnitude of the eigenvector approaches zero when the commanded 
maneuver approaches 180 deg. When cos 0 is sufficiently close to -1.0, 
the RL vector is replaced by a unit- commanded body axis according to the 
following strategy. 

The largest diagonal element of the BG matrix is determined. 


- 32 - 



Then, if 


BG largest, R L. = IL 
1 1 

BG,_ largest, RL _ U 

U L* M < 


BG,, largest, lOl = U 
33 t,a 


The above procedure ensures that the initial maneuver away from 
the 180-deg region is not made about an axis which is perpendicular to the 
desired maneuver axis. 

RL is considered the unit command rotation vector, and the 


variables dU^^ and dU^ are computed as 


dU f j = (RL X U^ a ) MLIM 


dU = (RLX U ) MLIM 
PL ' Pa/ 


and the interpolated body axes, on a minor cycle basis, are computed as 
described in Eqs. (54) and (55). MLIM is the magnitude of the rate limit. 

The method used to interpolate the commanded-body vector throughout 
the major cycle results in a very small nonorthonormality in the commanded 
vehicle axes. To prevent this nonorthonormality from growing, one can per- 
form an orthonormalizing process for each major cycle, as follows: 

Lu = + dTJ ei. )/llJ u + d LiJ 

L,a =1 ha X % a *«O nL ,/|U 5a X (U, a + dU nL )l (64) 

«na = % X V 


This process resets the command- vehicle coordinates to orthonormality 
every major cycle. 


- 33 - 


c. 


NAVIGATION 


The- guidance navigation calculations compute the 'present vehicle 
position and velocity using as inputs accumulated incremental velocity pulses 
from a triad of force rebalance integrating accelerometers. One of the 
accelerometers (WC1 is mounted on the IV U stable platform (turret); the 
other two (l;C and VC) are mounted on a platform that rotates, or carousels, 
at I rpm. Figure 4 >s a block diagram of the major and minor cycle 
navigation . 

1. ACCEL fill O V 1 1 T E R RESOLUTION AND COMPENSATION 


For the navigation function to he performed, the initial azimuth 
(at "go inertial") of the rotating platform must be determined. Further, 
whenever the incremental velocities are sampled, this azimuth must be 
updated. 

Four coordinate systems are used in the accelerometer resolution 
and compensation: 


[uc, vc, wcj 


[l,2,3] 


[u, v,w] 


[x,y, z ] 


Actual ca rouselling accelerometer input axes 
coordinate system. The coordinate system is, in 
general, nonorthogonal and rotating due to 1MU drift 
and ca rouselling. 

Ideal carouselling coordinate system This coordi_- 
nate system of convenience is defined as follows. 3 
is colinear with the carouselling axis of rotation. 

2 is perpendicular to 3; it is in_the plane of 3 and the 
vc accelerometer input axis. 1 completes the 
orthogonal right-hand set. 

Drifted launch site coordinate system. Initially 
equal to the initial vehicle roll, pitch, yaw coordi- 
nate system. After go inertial, related to the initial 
vehicle coordinate system by the IMU drift matrix. 

Earth centered inertial coordinate system. Defined 
as follows: 

x = unit vector in the equatorial and prime meridian 
planes at go-inertial time. 

z = unit earth spin vector, i.e., North Pole 

y - z x x to complete a right-hand set. 


34 





Figure 4. Navigation Block Diagram 










An miii.il > 1 /. 1 ‘nuih, or phase angle, is computed in the ground-in- 
flight interface program as follows: 


FANG - HTHYN + CFANG + CZZANG 


( 65 ) 


where 


HTHYN = the angle, computed in the ground alignment program, from 
north to the VC accelerometer input axis plus gyro mis- 
alignment errors 

CZZANG = gyro misalignment IMU compensation parameter 

CFANG = parameter to convert the phase angle from northeast to 

dov/nrange-crossrange coordinates and to time synchronize 
with the accelerometer readouts. 

The sine and cosine of the initial phase angle are then taken for resolution 
purposes . 


SS = sin(FANG) 
BS = cos (FANG) 


( 66 ) 


After initialization, the carousel phase angle sine and cosine is 
updated every 40 msec by a trigonometric identity as follows: 


SS n+l = < SS n Hcos 0.24°) + (BS )(sin 0.24°) 
BS n+1 = (BS n )(cos 0.24°) - (SSJisin 0.24°) 


(67) 


where 0.24 deg is the angle : ’car luselled" through in 40 msec. Also, the 
carousel phase angle is reinitiaJ zed to SS and BS at 1 -minute intervals. 

The accelerometer incremental counts are modified with the calibra- 
tion constants determined as part of factory calibration and pad test procedure 
The 40-msec frequency accelerometer compensation equations are 




AV uc = (CKAX)(AN - CKBXG) + (CKNX)(AN ) 2 

uc 

A ^vc (CKAY)(AN vc - CKBYG) + (CKNY)(AN v( J 2 

AV wc = (HZAF )(AN - HBZ) + (CKNZ)(AN ) 2 

wc 

whe re 


AN uc’ AN vc’ AN wc the raw counts from the carouselling uc and vc 

and the stationary wc accelerometers over the 
last 40-msec interval 

CKBXG, CKBYG = the uc and vc accelerometer factory bias 

calibration 


HBZ - ^e wc accelerometer bias calibration computed 
during final align 

CKAX, CKAY = the uc and vc factory scale factor calibration 

HZAF = the wc accelerometer scale factor calibration 
computed during final align 


CKNX, CKNY, CKNZ = the uc, vc, 


and wc nonlinearity calibration 


Continuing, one finds that 


ST1 = AV uc " ( czzb 2U)(AV ) - (CZZB3U)(A V t 

vc wc’ 

ST2 = AV - (CZZB3V)(AV ) 

VC WC ' 


(69) 


Equation (69) shows small angle approximations to misalignment 
rotation compensation where 


ST1 , ST2 = velocity increments in 1,1,3 coordinates 

CZZB2U - misalignment of the u axis toward ~Z 

CZZB3U = misalignment of the u axis toward 3 

CZZB3V = misalignment of tne v axis toward 3 


- 37 - 


ST1 and ST2 are then resolved into u,v,w coordinates as follows: 



IBl 






. 




mm 


AV um = (BS HST1) - (SS)(ST2) 


AV vm = (SS)(ST1) + (BS)(ST2) 


The resolved minor cycle velocity increments are finally 
accumulated as follows: 


A V <- AV A V 
up up um 


AV AV + AV 
vp vp vm 


[T 


J I 

■ *>W'! * 


A V f AV + A V 
wap wap wc 


Once per second, the accumulated velocity increments are sampled 
for use as inputs to the major cycle navigation equations 




AV 

«- AV 

u 

up 

AV 

*- AV 

V 

vp 

AV 

«- AV 

wa 

wap 

and AV 

are chei 


up vp wap 

further accumulation in the next major cycle interval. 


AV 0 
up 


AV «- 0 
vp 


AV <- C 
wap 




The w-accelerometer misalignments are performed on a major 
cycle basis, since the w instruments do not carousel. These misalignments 
do rotate, because the w-accelerometer is rotating at earth rate at liftoff. 
During guidance initialization, misalignments are computed in the u, v, w 
coordinate system as a function of the initial gimbal angle 


bWl = (CKB4W)(sin aj + (CKB5W)(cos a) 


BW2 = - (CKB4W)(cos a.) + (CKB5W)(sin a.) 


where 


BWl - w-accelerometer misalignment away from u axis 

BW2 - w-accelercmeter misalignment away from v axis 

CKB4W, CKB5W = factory-calibrated w-accelerometer misalignments 

in an a gimbal-oriented coordinate system 


The w-accelerometer input axis also "cones" after go inertial as a 
function of the inner gimbal angle, because of misalignment of the inner 
gimbal axis and the carouselling axis. The complete equation for 
w-accelerometer misalignment compensation is given in Eq. (75): 


AV = A V + [BWl + (CKBSW)(cos a) - (CKBCW)(sin a)]AV 
w w & u. 

+ [BW2 - (CKBCWMcos a) - (CKBSW)(sin »)]AV 


CKBSW, CKBCW = calibrated carouselling axis misalignments in an 

a gimbal angle coordinate system 


- 39 - 


2 . 


GYRO DRIFT COMPENSATION 

Gyro drift compensation is accomplished by computing the platform 
drift rate with calibration values obtained during hangar and pad tests. Then 
by time -integration of the drift rate, a matrix p can be computed that trans- 
forms the present (u,v,w) axes to the initial nondrifting axes. Further 
matrix operations then transform to the desired ECI coordinate system. 


rAV l 


rAV -i 

sx 


u 

AV 

sy 

= [CG] r [4>] T [p] 

AV 

V 

AV 

L sz J 


AV 

L w_ 


( 76 ) 


where 


AV 

sx, sy, sz 


measured (sensed) values of velocity increments in the 
ECI coordinate system 


AV 

u, v, w 


measured value of velocity increments along the, present 
drifted launch site coordinate system 


p - d rift matrix 

4* row rearrangement matrix 

CG - coordinate transformation matrix from LCI to ECI 
coordinates 


Note that p, cp and CG are shared by guidance (Paragraph III.B.6) and 
navigation. 

If the instantaneous drift rate vector is given as (p u> p v , p w ), the 
derivative of each column of the p matrix is the cross-product of the drift 
rate vector with that column vector. Calculating the cross products and 
combining terms, one obtains the matrix form 


p = [p] 


0 

6 

\ 


w 


-p 

0 

< P , 


w 


-p 


u 


0 


( 77 ) 


Integration of p is accomplished by using a second order Runge- 
Kutta algorithm. First, the p matrix is integrated a half cycle with an 
initial derivative: 


K~~ *n + < At/2) * n (78) 

The drift rate is then averaged over two cycles: 

A p = (A p + A p , , ) / 2 
r u r un r un+l 

A p = (A p + Ap , . ) / 2 ( < ^) 

r v r vn r vn+l' 

A? = (A p + A 0 , , )/2 

T w r wn r wn+l' 


A final step advances p as follows: 


n+1 


1 P n + 


[p 1 
L V n , 


0 

A ^ 

-Ap 

r w 


Ap 

r w 

0 

-A? u 

-Ap 

r v 

Ap 
y u 

0 


(80) 


Computation of the drift vector ip in u, v,w coordinates is complicated 
by the rotation of the carouselling instruments. Drifts of each gyro must be 
located spatially due to the arbitrary location of the 1MU turret and the 
rotation of the platform at 1 rpm. 

All drift matrix calculations are performed once per second. A 
central assumption for the drift calculations is that the drift occurred approxi- 
mately centered in the 1 sec compute cycle. To locate the carouselling 
instruments, one must compute backed-up values of the variables BS and SS. 


SSB = -(BS)(sin 3°) + (SS)(cos 3°) 


BSD - (BS)(cos 3°) + (SS){sm 3°) 


( 81 ) 


The variables SSB and BSB are then used to resolve AV and AV into 

u v 

coordinates representing the "average" position of the carouselling gyros 
over the last second. 


AVgi = (AV u )(BSB) + (AV v )(SSB) 


AV g2 = -(A~V u )(SSB) + (AV v )(BSB) 


(82) 


The fixed-torque and unbalance drifts of the carouselling gyros are then 
computed as 

DRFTU FT DU + (CKU1)(AV t ) + (CKU2)(AV g2 ) 

DRFTV = FTDV + (CKU3)(AV g2 ) + (CKU4)(AV gJ ) 


(83) 


where 

FTDU, V = fixed-torque drifts 
CKU 1 , 3 = spin-axis unbalance drifts 
CKU2, 4 = input-axis unbalance drifts 


In addition to the usual error sources associated with an IMU, 
carouselling itself introduces gyro drifts. The compensable ones are called 
gimbal- oriented bias (GOB), turret-oriented bias (TOB), and turret-oriented 
eta (TOE). GOB is a drift in the carouselling plane, which rotates as the inner 
gimbal rotates; i. e. , a drift fixed to the inner gimbal. Initially, GOB is 
computed in launch-centered coordinates as 


- 42 - 



GOBC1 = (CK20)(cos CK21) 
GCBC2 = (CK20)(sin CK2.1) 


( 84 ) 


where 


CK20 - GOB magnitude 
CK21 = GOB phase angle 

7 

Later, in flight, the computation of GOB reflects the rotation about 
the inner gimbal as follows: 


GOBI - - (GOB C2)(cos a) + (GOB Cl)(sin a) 
GOB2 = (GOB Cl )(cos a) + (GOB C2)(sin a) 


(85) 


I urret- or tented bias is a drift of the tui ret (perpendicular to the 
carouselling plane) that is a function of several harmonics of the turret present 
position with respect to the inner gimbal. To locate the turret during 
guidance initialization, one must compute the angle of the turret revolution 
from a zero inner gimbal angle value: 

- tan (sin Q-. /cos a. ) 

where 


°' 1 - initi al (at go inertial) inner gimbal angle 
sin a. = sine of initial a 
cos or. = cosine of initial a 


- 43 - 


Then, in flight, this a,ngle is updated as 




- tan (sin a ! cos a) + 


( 87 ) 


The equation for TOB is 


4 

TOB = [cos (n a - 0 ) 

f— { i L 'it l 

i=i 


( 88 ) 


In Eq. (88), n. represents integers specifying the harmonics of a ^ to 
be included in the TOB calculation,, 6^ represents the phase angle of each 
harmonic; and K represents magnitudes. 

TOE is a drift in the carouselling plane that is a function of the 
location of the turret with respect to the launch- centered coordinate system. 
It is computed at guidance initialization as 


TOE 1 = - (CK22)[ (sin a.)(cos CK23) + (cos *.)(sin CK23) J 
TOE2 = (CK22)l(cos m)(cos CK23) - (sin a.)(sin CK23) ] 


(89) 


who re 


CK22 TOE bias drift magnitude 
CK23 TOE bias drift phase angle 

Finally, the derivative of <p is computed as shown in Eq. (90): 


A p u = (BSBKDRFTU) - (SSB)(DRFTV) + GOBI + TOE1 
Ap v = (SSBMDRFTU) + (BSB)(DRFTV) + GOB2 + TOE2 (90) 

A p w = HRZ + (DIAU)(AV ) - (CZZU6)[ (AV v )(sin a.) + (AVJfcos ».)] + TOB 


- 44 - 


V 


whe "e 


3. 


BSB, SSB = sine and cosine of the average carousel angle 
during last compute cycle 

DRFTU, DRFTV = fixed-torque and g-sensitive drifts of the UC and 

VC carouselling gyros 

GOBI, GOB 2 = GOB drift [Eq. (84)] 

TOE 1 , TOE 2 = TOE drift [Eq. (89)] 

TOB = TOB drift [Eq. (88)] 

HRZ = noncarouselling Z gyro fixed-torque drift 

DIAU - spin-axis unbalance drift 

CZZU6 = input-axis unbalance drift, Z gyro 

TRANSFORMATION OF VELOCITY TO INERTIAL x,y,z 
COORDINATE SYSTEM 


The inertial x, y,z coordinate system is defined as follows: 

x = unit vector in the equatorial and prime meridian planes at 
go inertial time 

z = unit earth spin vector, i. e. , North Pole 
y = z X x to complete a right-hand set 


4. VEHICLE INERTIAL POSITION 

The following trapezoidal integration formulas are used to obtain 
the vehicle position (X, Y, Z) in the inertial coordinate system: 

X <■ X + V (At) + (1/2) (At) 2 (AV + AV ) 

Y «- Y + V (At) + (1/2) (At) 2 (AV + AV ) (9D 

y y o y 

Z Z + V (At) + (1/2) (At) 2 (AV + AV ) 

Z b Z gZ 


- 45 . 







where 


V , V , V 
x y z 


At 


AV , A V 
sx 

AV , A V 

g x 


sy 


gy 


» A V 


sz 


,AV 


gz 


x, y, z components of vehicle velocity 


major compute cycle interval - 1 second 


x,y,z components of sensed vehicle velocity 
increment from Eq. (76) 

x, y, z components of velocity increments due to 
gravity 


Vehicle position magnitude is then calculated as 

2 2 2 1/2 

r = (x + y + z ) 


(92) 


5. GRAVITY COMPUTATIONS 

An approximation for the gravitational potential of the earth is 

U 

U = (n/R)[l + (Ja 2 /3R 2 )(1 - 3 sin 2 \) ] (93) 

where 


p = gravitational parameter of the earth 
R = distance from center of the earth 
\ - latitude 

a = equatorial radius of the earth 

J = a characteristic constant that is a function of the moments of 
inertia with respect to the polar and equatorial axes 

A derivation of Eq. (93) is contained in Ref. 7. This approximation, 
which is a truncated series, contributes negligible inaccuracies to the overall 
navigation function. 


- 46 - 


Then, from the definition of the Z component of vehicle position, 
it follows that 

sin 2 \ = 2 2 /R 2 (94) 

which, upon substitution into Eq. (93), yields 

U = (p/R) (1 + Ja ? '/3R 2 - Ja 2 Z 2 /R 4 ) (95) 

The x, y, z components of the acceleration due to gravity are then 
obtained by the partial differentiation of Eq. (95) with respect to each axis, 
and 

g x = -aU/9x = -(8U/9R)OR/ax) 

gy = -au/a y = (-au/aR)OR/a y ) (96) 

g = -au/az = (-au/aR)OR/az) 

The negative sign is added by convention. 

An evaluation of the common term 3U/aR yields 

aU/aR = (-p/R 2 )(l + Ja 2 /R 2 - 5Ja 2 Z 2 /R 4 ) (97) 

2 

and, by factoring out - -(p/R ), one obtains 

aU/3R = A (1 - Ja 2 A /p + 5 Jr. 2 A Z 2 /pR 2 ) (98) 

8 8 8 


- 47 - 



In addition, one finds that 


dR/dx = X/R 
9R/9y = Y/R 
dR/dx = Z/R 


(99) 


and 


aU/az = ZfxJa i 2 Z/R 5 = +2Ja Z A 2 Z/|j.R 

8 


( 100 ) 


The velocity increments over a major compute cycle due to gravi- 
tational acceleration arc given in Eq. (101). 


AV gx g x -(ou/aR)(aR/ax) 


a v g - -(au/aR)(aR/ax) 

Bi y 


AV =g - -OU/9R)OR/8z) 

gz b z 


6. VEHICLE INERTIAL VELOCITY 

The inertial velocity of the vehicle is computed as 


( 101 ) 


(V 


.). (V) + (1 /2)[(AV .) , + (AV .) ] [At] + (AV )(At) 

in) 1 in L gi n+1 gi n J 1 J si 


i = x,y, z 


gi 


( 102 ) 


Averaging AV over two cycles produces a trapezoidal integration of acceler- 


D. ATTITUDE ERRORS 

Attitude errors art? < rucial outputs of the guidance equations utilized 
by the Digital Flight Controls Equation in stabilizing the vehicle to a desired 
attitude. They represent differences between the present actual vehicle 
attitude and the present desired vehicle attitude. The desired attitude is 
computed by the guidance maneuvering equations (Paragraph III. B. 6); the 
actual attitude is derived from synchros attached to each gimbal. 


1 . 


form 


SYNCHROS 

The synchro signals supplied to the inflight computer are of the 


VI = K sin (0-120°) and V2 = K sin (0-60°) 


(103) 


where G can be any of three gimbal angles, a, p, or y„, and K is a scaling 
factor common to VI and V2. 

By trigonometric identities, one obtains 

VI - V2 = K sin (0-120°) - K sin (8-60°) 


= K[(sin 0 cos 120° - cos 0 sin 120°) 
- (sin 6 cos 60° - cos 0 sin 60°)] 

= K sin 0 (cos 120° - cos 60°) 


(104) 


-K sin 8 


- 49 - 


a n<l 

VI I V2 K sin (0 - 120°) + K sin (0 - 60°) 

K | (sin 0 cos 120° - cos 0 sin 120°) 

+ (sin 0 cos 60° - cos 0 sin 60°) | 

(105) 

- K cos 0 (- sin 120° - sin 60°) 

= -K*/3 cos 0 

Thus, by simple sum and difference, one obtains the sine and 
cosine of the gimbal angles, except for a common multiplier -K and the 
constant V 3. 

Since the sine and cosine of the gimbal angles are desired on a 
minor cycle basis as a convenience for the computation of vehicle attitude 
errors, it is necessary to determine the scaling factor K. 

At go inertial, the following initial calibration is performed for 
each of the three gimbal angles: 

X - V2 - VI 
Y = (VI + V2)(C1) 

(106) 

K 2 - (X)(X) + ( Y)( Y) 

K -v/ K 2 

In this series of equations. Cl is a constant equal to -1.0/\f3, and K 
is the calibrated value of the synchro scale factor. 

On succeeding minor cycles, the sine and cosine of each gimbal 
angle are computed as follows: 


- 50 - 



X = V2 - VI 
m 

Y m = < V1 + V2 > (C1) 

K 2 - (X )(X ) + (Y )(Y ) 

m m m m m 

K - (1/2) fK 2 /K + K ) 
m m m rn 


107 ) 


sin 0 = X/K 
cos 0 = Y/K 


m 


m 


These minor cycle computations use Newton’s square-root algorithms, 
which converge quite rapidly if the initial value is relatively accurate. 

Since the three gimbals, n, (3, and y^, are not necessarily zero at 
go inertial, they are read initially; the initial values are used to compute the 
three angles through which the vehicle rotates during the flight. Taking a as 
an example, trigonometric identities yield 


sin a = sm a cos a. - cos a sin a 
pi pi 


cos a = cos a cos a. + sin a sin a 
pi pi 


(108) 


where 


sin a, cos a = sine and cosine of vehicle angular rotation 
sin a , cos a = present gimbal synchro sine and cosine readings 
sin o , cos m = initial gimbal synchro sine and cosine readings 


An identical procedure is followed to compute the sine and cosine of the }3 
and v D vehicle rotations. 


- 51 - 


The fourth gimbal y has very limited travel (‘"10 deg), and the 
following simplifying approximations are made: 

cos y - 1.0 


( 109 ) 


sin y ~ (CKDSCA}(sin 'y ) - sin y^ 


where CKDSCA = calibrated scale factor for sin 


ATTITUDE ERROR COMPUTATIONS 


The IMU gimbal angles (a, p, Y and Y^) ideally establish the 
relationship between two vehicle coordinate systems: the present roll, 
pitch, and yaw (£,q,p body axes; and the initial body axes (£.,C,£.l. Initial 
alignment is such that, at launch, a positive roll results in a negative gimbal 
angle «; a positive yaw results in a positive gimbal angle (3; and a positive 
pit< h results in a positive gimbal angle Y^. The fourth gimbal Y, in normal 
circumstances, is zeroed. Its sign convention is the same as Y 

i\. 


[uc.r,|.u, 


CO* tp 0 -sin 

0 I 0 

\ K <» * o« \ 


(' sin P o i » ■<*> > 

i 

- sin ,1 ( ,1 1(1 (j 

I I 


ljr , 0 




.] (110) 


ij si n it i 06 


I 06 / | 


Performing the matrix multiplication with cos y ~ 1.0 yields 


- 52 - 







(' '>■> 1 1 ( , ‘ " i“i ) l( --'ii 'i ) 

(» ■ >& \ ^ sin 6 cos u 

( - cos ^ ^ sin >i sin / 


- cos cos 0 sin \ sin a 

- cos y^ c ob 3 sm \ cos a 


- -nn ^ sin o) 

- sin Yp cos <>) 

(- bin |i) 

(t os 3 cos o + 

( - cos (1 sin a + sin 3 sin \ tos 


sir* 3 sm y s in a 


(sin > ^ cos il t cos y ^ sin y 

(sin Y{^ sin 3 cos a 

(- sin y ^ sin 3 sin u 


- sin cos p sin y sin o 

- 6in y^ cos 3 sin y cos a 


+ cos sin ») 

f cos y . Cos o) 


:n i) 


or M = lUg, U^, U^j 


If the matrix of initial vehicle axes is assumed to be identity, the 
rows of the M matrix comprise the present measured vehicle axes in a 
gimbal-oriented coordinate system (a,b,g), where initially 


a = Uc 


b = U 


ni 




( 112 ) 


From the actual vehicle axes and the present desired vehicle axes, attitude 
errors can now be computed. 

Consider a three-axis (roll, pitch, yaw) maneuver to reorient the 
vehicle from its actual attitude to the desired attitude, as three Euler 
rotations in the following order: roll, pitch, yaw. In matrix form, tins is 


- 53 - 




0 


0 


B O cos R sin R 


cos P 0 -sin P I f cos Y sin Y 0 ~J 


-sin Y cos Y 0 


0 - sin R cos R sin P 0 cos P 


where R, P, and Y are the respective roll, pitch, and yaw rotations. 
The matrix multiplication yields 


(113) 


cos P cos Y 


-cos P sm Y 


sin P 


sm R sin P cos Y 
+ cos R sm Y 

sm R sin P sin Y 
+ cos R cos Y 

-sin R cos P 


•cos R sin P cos Y 
+ sm R sin Y 

cos R sin P sin Y 
+ sm R cos Y 

cos R cos P 


(114) 


Simplifying B with small angle approximations yields 


"1 Y -P 

B« -Y 1 R 
P -R 1 


(115) 


Another expression for B in terms of body axes is 




§dm 

- r, 5 

^ £dm 

• U 

n 

Ut . 

§dm 

• U £ 


> 

B - 

0 . 
pdri) 

'°£ 

0 . 
i|dm 

• U 

n 

U . 
r)dm 


(116) 



%m 


f,dm 

• U 

n 

i 

t,dm 

• D J 






REFERENCES 


1 E. Da maud , D. L. Kleinbub, and J. B. Shaul, Guidance, Control 
and Ground Equations for SSLV C-26 , Aerospace Corporation Report 
No. TOR- 01 7 2(21 { 2- 02)- 1 1 , Reissue B (1 November 1972), Vol. I. 

Program 624A Mission Specification for Flight Plan VII- J , 

Aerospace Corporation Report No. TOR-01 72(21 1 2- 02)-6 , Rev. 1 
(28 September 1972). 

H. Sokoloff, Program 624A Discrete List for Flight Plan VII- J , 
Aerospace Corporation Report No. TOR-0172(21 12-02)-l6, 

Reissue A (24 November 1972). 

George W. Cherry, "A General Explicit Optimizing Guidance Law 
for Rocket Propelled Spacecraft, 11 Proceedings of the AIAA/ION 
Astrodynamics Guidance and Control Conference , AIAA Paper 
No. 64-638 (24-26 August 1964). 

D. MacPhcrson, An Explicit Solution to the Powered Flight 
Dynamics of a Rocket Vehicle , Aerospace Corporation Report No. 
TOR- 1 69(31 26)TN-2 (31 October 1962). 

C. W. Pittman, The Design of Explicit Guidance Equations for 
Rocket Ascent, Aerospace Corportation Report No. 

T DR- 469(5 5 40 - 1 0)- 4 (24 May 1965). 

F.E. Darnaud, A Technique Allowing Continuous Operation of 
Integral Control , Aerospace Corporation Report No. 

TO R- 1 00 1 (2 1 1 6 -60)- 8 (1 3 September 1966). 

K. A. Ehricke, "Environmental and Celestial Mechanics," 

Principles of Guided Missile Design, Vol. I: Space Flight 
(D. Van Nostrand Company, Inc., Princeton, N. J. , I960). 

Carousel V Inertial Navigation System, System Technical Description 
EP0137, Delco Electronics Division of General Motors, Milwaukee, 
Wisconsin, 1969. 


- 57 - 


PRECPDING PAGE BLANK-HOT FI